We study Anderson localization on high-dimensional graphs with spatial structure induced by long-ranged but distance-dependent hopping. To this end, we introduce a class of models that interpolate between the short-range Anderson model on a random regular graph and fully connected models with statistically uniform hopping, by embedding a random regular graph into a complete graph and allowing hopping amplitudes to decay exponentially with graph distance. The competition between the exponentially growing number of neighbors with graph distance and the exponentially decaying hopping amplitude positions our models effectively as power-law hopping generalization of the Anderson model on random regular graphs. Using a combination of numerical exact diagonalization and analytical renormalized perturbation theory, we establish the resulting localization phase diagram emerging from the interplay of the length scale associated to the hopping range and the onsite disorder strength. We find that increasing the hopping range shifts the localization transition to stronger disorder, and that beyond a critical range the localized phase ceases to exist even at arbitrarily strong disorder. Our results indicate a direct Anderson transition between delocalized and localized phases, with no evidence for an intervening multifractal phase, for both deterministic and random-hopping models. A scaling analysis based on inverse participation ratios reveals behavior consistent with a Kosterlitz-Thouless–type transition with two-parameter scaling, in line with Anderson transitions on high-dimensional graphs. We also observe distinct critical behavior in average and typical correlation functions, reflecting the different scaling properties of generalized inverse participation ratios.

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• Anderson localization
• Disordered systems
• Random graphs
• Exact diagonalization
• Green's function methods
• Perturbative methods